Functions and algebra: Investigate and use instantaneous rate of change

• Determine limits of functions intuitively.
• Distinguish between the value of a function at a particular point and the limit of that function at that point.
• Establish the derivatives of the following functions from first principles:
  [latex]\scriptsize f\left( x \right)=b,~~f\left( x \right)=x,~~f\left( x \right)=ax+b,~~f\left( x \right)={{x}^{2}},~~f\left( x \right)=a{{x}^{2}}+b[/latex]
• Find the derivative of functions in the form:
  • [latex]\scriptsize \displaystyle f(x)=a{{x}^{n}}\text{or }y=a{{x}^{n}},{f}'(x)=n.a{{x}^{{(n-1)}}}\text{or }\displaystyle \frac{{dy}}{{dx}}=n.a{{x}^{{(n-1)}}}[/latex] Examples to include: [latex]\scriptsize 4{{x}^{2}};~\displaystyle \frac{3}{{{{x}^{{-3}}}}};\displaystyle \frac{2}{{\sqrt[3]{{{{x}^{2}}}}}};\displaystyle \frac{5}{{3{{x}^{2}}}}[/latex] (all examples within this range).
• Use the constant, sum and/or difference rule by first simplifying the expression.
  • If [latex]\scriptsize y=f\left( x \right)=a~[/latex] and [latex]\scriptsize a[/latex]is a constant function then: [latex]\scriptsize \displaystyle \frac{{dy}}{{dx}}={f}'\left( x \right)=0~[/latex]
  • If [latex]\scriptsize y=\text{k}f\left( x \right)\text{ }\!\!~\!\!\text{ }~[/latex]then [latex]\scriptsize \displaystyle \frac{d}{{dx}}\left[ {\text{k}f\left( x \right)} \right]=\text{k}\displaystyle \frac{d}{{dx}}\left[ {f\left( x \right)} \right][/latex]
  • If [latex]\scriptsize y=f\left( x \right)\pm g\left( x \right)~~[/latex]then [latex]\scriptsize \displaystyle \frac{d}{{dx}}\left[ {f\left( x \right)\pm g\left( x \right)} \right]=\displaystyle \frac{d}{{dx}}\left[ {f\left( x \right)\left] {\pm \displaystyle \frac{d}{{dx}}} \right[g\left( x \right)} \right][/latex].
• Solve maxima and minima problems about real life situations from given equations.

By the end of this unit you will be able to:

• Determine limits of functions intuitively.
• Distinguish between the value of a function at a particular point and the limit of the function at that point.

By the end of this unit you will be able to:

• Develop an intuitive understanding of the gradient at a point.
• Differentiate using first principles.

By the end of this unit you will be able to:

• Apply the constant, sum and difference rules for differentiation.
• Use notation appropriately.

By the end of this unit you will be able to:

• Use differentiation to find optimal solutions.